Biaxial tensile behavior of stainless steel 316L manufactured by selective laser melting

In this study, miniaturized cruciform biaxial tensile specimens were optimized by finite element simulation software Ansys to vary five geometric parameters. The optimized specimens were utilized to characterize the biaxial tensile properties of 316L stainless steel fabricated through selective laser melting (SLM), with the two loading directions being vertical (X) and parallel (Y) to the building direction. It was discovered that at load ratios of 4:2 and 2:4, the yield strengths along X and Y orientations reached their respective maxima. By comparing the experimentally obtained yield loci against predictions by theoretical criteria including Mises, Hill48 and Hosford, it was found that the Hill48 anisotropic criterion corresponded most closely with the experimental results, while the other two criteria exhibited considerably larger deviations. Therefore, Hill48 was concluded to most accurately describe the yielding behaviors of SLM 316L under complex loading conditions.

the most commonly adopted sample geometries for biaxial tensile experiments due to its structural symmetry and relatively uniform stress state in the center gauge area 29,30 .The dimensions of a cruciform specimen need to be carefully designed to achieve a balanced biaxial stress state during testing [31][32][33] .Common design factors include the sample length, width, thickness, fillet radius, and length of gauge area.Proper dimension ratios have been analytically derived and experimentally validated to minimize undesirable stress concentrations for various materials 31,[34][35][36] .However, optimized geometries often have large volumes, which increases material waste.Therefore, this study utilizes finite element software Ansys to simulate biaxial tension experiments for designing miniaturized cruciform specimens and optimizing their geometric parameters.
Accurately predicting the yielding behaviors of metallic materials is essential for simulating their plastic deformation in manufacturing processes 37,38 .The von Mises, Hill 1948 (Hill48) and Hosford yield criteria are among the most established anisotropic yield functions suitable for various metals [39][40][41][42][43] .The von Mises criterion assumes that yielding occurs when the von Mises stress reaches a critical value, applicable for isotropic metals like mild steel 44 .The Hill48 criterion introduces anisotropy by using different yield stresses along three material directions, providing good accuracy for cubic metals 37,45 .The Hosford criterion generalizes the isotropic von Mises model with a tunable exponent parameter.The exponent value can be fitted to experimental data to capture anisotropic yielding.Compared to Hill48, Hosford yield surfaces have smoother corners better matching measured shapes.Two general yield criteria, isotropic Mises and anisotropic Hill48, are included in commercial nonlinear finite element software ABAQUS.The accuracy of the yield criterion can be verified by the results of biaxial tensile test.
In this study, miniaturized cruciform biaxial tensile specimens was designed and optimized by the finite element software Ansys.The biaxial mechanical behavior of SLM 316L was studied by the biaxial tensile test with the optimized miniaturized cruciform specimens, and the stress-strain curves under different load ratios were obtained.The yield loci of SLM 316L under biaxial stress is obtained by calculation.The yield loci obtained by experiments were compared with those calculated by Mises, Hill48 and Hosford yield criteria to verify the accuracy of different yield criteria.This can provide a reference for the numerical simulation to predict the performance of parts.

Design optimization of miniaturized cruciform specimens
To conduct biaxial tensile tests, cruciform specimens were designed using finite element analysis software ANSYS.Biaxial tensile simulations were performed to obtain the stress states of the specimens, based on which the geometries were optimized.Tensile tests employed cruciform specimens with overall dimensions shown in Fig. 1.Five geometric factors of the specimens were optimized, including thickness of the center gage section, width of the straight arm notches, length of the straight arm notches, number of notches, and fillet radius of the inner corner.The levels of each factor are listed in Table 1.Through iterative simulation and optimization, a cruciform specimen design inducing balanced biaxial stresses was obtained.The adopted specimen geometry   and dimensions enabled reliable characterization of the plastic deformation behavior through subsequent biaxial tensile experiments.A L 15 (4 5 ) orthogonal array was implemented for the orthogonal experimental design.Owing to the cruciform symmetry, only a quarter fraction was modeled for finite element analysis, as illustrated in Fig. 2 showing the FE model of the specimen.

Materials and additive manufacturing
The SLM process utilized a commercial gas-atomized 316L powder with particle size distribution shown in Table 2 and nominal composition listed in Table 3. Printing occurred under argon atmosphere to prevent oxidation.The adopted processing parameters are provided in Table 4.A reciprocating scanning strategy with 67° rotation between layers was implemented.The directly fabricated cruciform specimen by SLM is depicted in Fig. 3.

Biaxial tension testing
In situ biaxial tensile tests of SLM 316L cruciform specimens were performed using an IPBF-5000 system (CARE Measurement and Control Co., Tianjin, China).Full-field surface strain measurements were obtained through     www.nature.com/scientificreports/

Design optimization of miniaturized cruciform specimens
This study optimized the cruciform specimen geometry focusing on achieving uniform stress distribution in the center gage section.The stress uniformity γ of the center gage was defined as: where m is the number of selected reference points (m = 6, Fig. 5), and σ i mises is the von Mises stress at each reference node.
Figure 5 illustrates the schematic of the reference nodes.Finite element models of cruciform specimens with dimensional variations per the factor levels in Table 1 and L15 orthogonal array were generated.Biaxial tensile simulations provided the stress states during loading.The stress uniformity γ was calculated using Eq. ( 1) based on the nodal stresses.Tables 5 and 6 list the computed γ values for node 1 stresses of 200 and 500 MPa, respectively. (Factors A, B, C, D, E in the table correspond to   www.nature.com/scientificreports/ the thickness of the center gage section, width of the straight arm notches, length of the straight arm notches, number of notches and fillet radius of the inner corner respectively).
The influence of each factor on γ was determined using the Statistica software based on the computed results in Tables 5 and 6, as depicted in Figs. 6 and 7.
Figure 6 indicates geometry A 2 B 2 C 3 D 2 E 1 conferring optimal γ under 200 MPa, while Fig. 7 shows geom- etry A 3 B 2 C 2 D 1 E 1 being optimal for 500 MPa.Although one notch (D 1 ) minimally influenced γ at 500 MPa, it exceeded the average effect at 200 MPa.Similarly, three notches (D 2 ) also surpassed the mean impact on γ at Table 6.Orthogonal array and results ( σ 1 mises = 500 MPa).www.nature.com/scientificreports/500 MPa.In contrast, five notches (D 3 ) exhibited relatively low effects at both stresses.Hence, five notches (D 3 ) were chosen for the notch number.Considering diminished formability with excessively small fillet radius, 0.25 mm (E 1 ) was replaced with 0.5 mm (E 2 ) .Likewise, the notch width was increased from 0.4 mm (B 2 ) to 0.8 mm (B 4 ) .Integrating these factors, geometry A 3 B 4 C 3 D 3 E 2 was selected as the optimum cruciform design, with final dimensions listed in Table 7.

True stress-true strain curves for different loading ratios
To obtain the true stress-true strain curves under nine loading ratios, the area of the mid-plane cross-section in the gage region was defined as S, calculated by: where l 0 , w 0 , t 0 are the initial length, width and thickness of the gage section.l , w , t represent the instantaneous counterparts, and ε is the strain along the measurement direction perpendicular to the cross-sectional area S.
The true stress σ can then be expressed as: Figure 8 presents the true stress-true strain curves of SLM 316L stainless steel under nine biaxial tension loading ratios.The flow behavior varied with the stress state, and the strain hardening exponent increased as the load ratio shifted from uniaxial toward balanced biaxial tension.Table 8 lists the yield strengths along the two directions under various loading ratios.Figure 9 presents the evolution of yield strengths along the two directions with varying loading ratios.The peaks were attained at 4:2 and 2:4 ratios for the respective orientations, differing from conventionally forged 316L that typically peaks at equivalent ratios 47 .

Experimental yield locus
To determine the yield points for the tensile tests, true plastic strains ε p x and ε p y along the X and Y orientations were calculated using Eq. ( 4).
where C x and C y are the slopes of the elastic portions of curves ε x − σ x and ε y − σ y measured in MPa from the biaxial tensile tests.ε x and ε y represent the true strains along the X and Y orientations, respectively.
For simplicity, plastic work contours are often considered equivalent to experimentally measured yield point trajectories 48 .Uniaxial tensile tests on SLM 316L vertical to the build direction using ASTM E8 specimens provided uniaxial true stresses σ p 0 corresponding to plastic strains ε p 0 of 0.002, 0.006 and 0.01.The plastic work W was measured per unit of plastic strain ε p 0 .In biaxial tensile tests with fixed stress ratios, the sum of plastic work along both orientations was obtained.Equivalent yield loci were identified when the unit plastic works W were equal under different stress states.For instance, as depicted in Fig. 10, point σ p * 1 , σ p * 2 represents a given biaxial tension stress state 49 .Point σ * , ε p * refers to the uniaxial stress-strain curve vertical to the building direction, satisfying: Notably, the integration in Eq. ( 5) was performed using discrete numerical integration, specifically integrating discrete trapezoidal areas encompassed by the data points and horizontal axis.This method yielded yield loci at three equivalent plastic strain levels for SLM 316L, as depicted in Fig. 11.
The yield contours demonstrate similar evolving trends in shape and size with increasing plastic deformation.Per the convexity principle, the plastic work contours expand outwards.Owing to strain hardening, the contours intensify from center to periphery at a given plastic strain increment.Notably, anisotropic mechanical properties induced asymmetry in the yield locus shapes along the balanced biaxial tension path, deviating from classic isotropic predictions.This signifies that deformation history and direction dependency in the SLM-processed 316L stainless steel influence yielding even at relatively small strains.The expanded yield area indicates enhanced formability, but the asymmetric distortions suggest potentially complex yielding characteristics under multiaxial loading.

Comparison and analysis of experimental and theoretical locus
This section compares the experimentally obtained yield loci against predictions by theoretical yield criteria (Mises, Hill48, Hosford).The calculation of the parameters in theoretical yield criterion is adequately elaborated in Refs. 40,43.To quantify the correspondence between the calculated and measured yield points, the mean error δ was defined as an accuracy metric:  www.nature.com/scientificreports/www.nature.com/scientificreports/where σ i 1 , σ i 2 denotes the experimental yield point coordinates, d i is the normal distance from the point to the calculated yield contour, and n is the number of experimental points.Figure 12 presents the measured yield loci versus those predicted by theoretical criteria.Figure 13 shows the mean errors between experimental and calculated values.
As shown in Fig. 12, the Hosford yield criteria deviated considerably from experiments under 4:2 and 2:4 loading ratios, with significant inaccuracies in balanced biaxial tension prediction.The Hosford yield criteria, grounded in crystalline plasticity principles, precludes shear stress components in its formulation.However, SLM 316L stainless steel possesses intricate grain morphological distributions, which the limited anisotropy

2 Figure 5 .
Figure 5.The schematic of the reference nodes.

2 Figure 8 .
Figure 8. True stress-true strain curve of SLM 316L for different loading ratios, (a) X direction, (b) Y direction.

Figure 9 .
Figure 9. Evolution of yield strength with loading ratios.

Table 1 .
Levels of geometric shape factors for cruciform specimens.

Table 2 .
The particle size distribution of powder.D10, D50 and D90 refer to the particle size corresponding to the accumulative distribution of 10, 50 and 90%, respectively.

Table 3 .
Chemical composition of 316L stainless steel powder.

Table 4 .
The processing parameters of SLM.

Table 7 .
Optimal geometry of cruciform specimens.

Table 8 .
Biaxial tensile tests data at various load ratios.